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Keldysh was born to a professional family of Russian nobility. His grandfather, Mikhail Fomich Keldysh (1839–1920), was a military physician, who retired with the military rank of a General. Keldysh's grandmother, Natalia Keldysh (née Brusilova), was a cousin of famous general Aleksei Brusilov. Keldysh's maternal grandfather, Alexander Nikolaevich Skvortsov, was a General of Infantry, a hero of the Caucasian War.

Several members of the Keldysh family were victims of political repressions. In the 1930s Keldysh's uncle was sent to a labor camp on the White Sea-Baltic Canal construction site. In 1935 Keldysh's mother was arrested but after a few weeks was released. It was a part of the campaign of collecting gold from the population, but after Keldysh's father brought all the jewelry the family had, the unsatisfied NKVD officer returned "all this garbage" back. Keldysh's brother Mikhail, a historian who specialized in Medieval Germany, was arrested in 1936 and executed in 1937 as a "German spy." In 1938 another of Keldysh's brothers, Alexander, was arrested as a "French spy." Alexander was spared because of a small liberalization of the repressions during the transfer of the NKVD leadership from Nikolai Yezhov to Lavrenty Beria—he was acquitted in the court.

Working in TsAGI he explained the auto-oscillation effects of flutter (in-flight auto-induced oscillations and structural deformations), and shimmy (auto-oscillation in the nose-wheel of aircraft undercarriages while on the ground). Both works were mathematically beautiful and practical; both were important as both effects were responsible for many aircraft catastrophes at the time.

In 1937 Keldysh became Doctor of Science (his dissertation's title was Complex Variable and Harmonic Functions Representation by Polynomial Series) and a Professor of Moscow State University. In 1943 he became a Corresponding Member of the USSR Academy of Sciences. He got his first Stalin Prize in 1946 for his works on aircraft auto-oscillations. In 1943 he also became a full member of the Academy and the Director of NII-1 (Research Institute number 1) of the Department of the Aviation Industry. He also headed the Department of Applied Mechanics of the Steklov Institute for Mathematics (in 1966 the department became Institute for Applied Mechanics, named after Keldysh).

During the 1940s Keldysh became the leader of a unique group of applied mathematicians involved in almost all large scientific projects of the Soviet Union. Keldysh created the Calculation Bureau that carried most of the mathematical problems related to the development of nuclear weapons. The bureau is also credited with design of the first Soviet computers. In 1947 he became a member of Communist Party.

In 1954 Keldysh, Sergey Korolyov and Mikhail Tikhonravov submitted a letter to the Soviet Government proposing development of an artificial satellite to orbit the Earth. This letter began the effort that culminated in the world's first satellite, Sputnik in October 1957, which marks the beginning of mankind's Space Age. In 1955 Keldysh was appointed chairman of the Satellite Committee at the Academy of Science. In recognition of his contribution to the problems of defense Keldysh was awarded the Hero of Socialist Labor (1956) and the Lenin Prize (1957). In 1961 he received a second Него of Socialist Labor medal for his contribution to Yuri Gagarin's flight into space, the first person to orbit the earth.